Question

Derivada Considere g(x)=(x²-5)³/(x²+4)², determine o valor
de g`(2)

Answer 1

$\boxed{g(x)= \frac{(x^2-5)^3}{(x^2+4)^2} }$

derivada do quociente
$\frac{U}{V} =\boxed{\boxed{ \frac{(U'*V) -(U*V')}{V^2} }}$

U =$U=(x^2-5)^3$

derivada de $(a)^n = n*(a)^{n-1} * (a')$

(a') é a derivada do que esta dentro do parenteses

aplicando isso
$U' = 3*(x^2-5)^2 *(2x)\\\\U'=6x(x^2-5)^2$
.................................................................................................
$\boxed{V=(x^2+4)^2}\\\\\ V'=2(x^2+4) *2x\\\\ V' =4x(x^2+4)\\\\\boxed{V'=4x^3+16x}$

a derivada fica
$\frac{6x(x^2-5)^2*(x^2+4)^2-(x^2-5)^3*(4x^3+16x)}{((x^2+4)^2 )^2} \\\\\\\\\boxed{\boxed{g'(x)= \frac{6x(x^2-5)^2*(x^2+4)^2-(x^2-5)^3*(4x^3+16x)}{(x^2+4)^4} }}$



agora calculando g'(2) 

$g'(2)=\frac{6*2(2^2-5)^2*(2^2+4)^2-(2^2-5)^3*(4*2^3+16*2)}{(2^2+4)^4} }\\\\\\g'(2)=\frac{12*(-1)^2*(8)^2-(-1)^3*(64)}{(8)^4} }\\\\\\\boxed{g'(2)= \frac{13}{64} }$